1 LOGO Chapter 1 Vector Algebra Part One iugaza2010.blogspot.com melasmer@gmail.com Vector Analysis Electrostatic fields Magnetostatic fields Electromagnetic fields(wave) 3 Unit Vector of a vector A is a vector whose magnitude is unity and its direction is along A 4 Example ___ A 3ax 4ay 5az A aA |A | Ax | A | 3 2 2 Ay Az 4 5 2 2 2 2 50 7.071 3ax 4ay 5az aA 7.071 aA 0.42 ax 0.5657 ay 0.7071 az | aA | 1 , , magnitude 1, , direction along A 5 Position Vector of point P is a vector directed from origin to P e.g. OP=P-O= 2 ax + 3 ay + 4 az 6 Distance Vector directed from one point to another point . e.g. PQ=Q-P= =(0-1)ax+(3-2)ay+(1-5)az =-ax + ay - 4 az 7 P1(2,4,4) , P2(-3,2,2) Find the unit vector along P1P2 P1P 2 P 2 P1 (3 2)ax ( 2 4) ay (2 4)az 5ax 2ay 2az P1P 2 5ax 2ay 2az a P1P 2 0.87ax 0.348ay 0.348az | P1P 2 | 25 4 4 8 A=ax+3az B=5ax+2ay-6az Find: (a) |A+B| | A B || 6ax 2ay 3az | 36 4 9 7 (c) The component of A along ay zero 9 (d) Unit vector parallel to 3A+B 3 A B 8ax 2ay 3az | 3 A B | 64 4 9 77 a3 A B 8ax 2ay 3az 0.91168ax 0.22799ay 0.3419az 77 10 Points P(1,-3,5) , Q(2,4,6) , R(0,3,8) Find: (a) Position vectors for P and R OP P O ax 3ay 5az OR R O 3ay 8az (b) Distance vector QR QR R Q (0 2)ax (3 4)ay (8 6)az 2ax ay 2az 11 Points P(1,-3,5) , Q(2,4,6) , R(0,3,8) Find: (c) Distance between Q and R QR 2ax ay 2az | QR | 4 1 4 3 12 Vector Multiplication (1) Dot Product (Scalar) A=Ax.ax+Ay.ay+Az.az B=Bx.ax+By.ay+Bz.az A.B=|A| |B| cos θ A.B=AxBx+AyBy+AzBz Note: A.B=-|A| |B| cos θ 13 Notes: (1) If A.B=0 θAB =90 Orthogonal ax.ay=0 ax.az=0 ay.az=0 2 (2) ax.ax=|ax| =1 2 ay.ay=|ay| =1 2 az.az=|az| =1 (3) 2 A.A=|A| |A| cos 0 =|A| 14 (2) Cross Product (Vector) A=Ax.ax+Ay.ay+Az.az B=Bx.ax+By.ay+Bz.az AxB=|A| |B| sin θ .an AxB= • Cross product is a vector direction: Orthogonal to A and B plane magnitude: area of parallelogram 15 متوازي االضالع Notes: (1) AxB=-(BxA) (2) ax x ay = az ay x az = ax az x ax = ay ax x az = -ay (1) AxA=|A| |A| sin 0 = 0 16 e.g. A=2ax-ay-2az B=4ax+3ay+2az )(AxB).A=|AxB||A|cos 90= 0 (normal )(AxB).B=0 (normal العمودً علي المستوى عمودً علي أً متجه يحتويه هذا المستوى 17 A=ax+3az B=5ax+2ay-6az Find the angle between vector A and B A.B=5+0-18=-13 | A | 1 9 10 | B | 25 4 36 65 A.B | A || B | cos 13 cos 0.5099 10 65 cos1 (0.5099) 120.65 18 A=2ax+ay-3az B=ay-az C=3ax+5ay+7az Find: 2 (d) A.C - |B| A.C ( 2 * 3) (1* 5) ( 3 * 7) 6 5 21 10 | B | 1 1 2 | B |2 2 A.C | B |2 10 2 12 19 A=2ax+ay-3az B=ay-az C=3ax+5ay+7az 1 1 1 (d) B x ( A C ) 2 3 4 1 1 1 B ay az 2 2 2 1 2 1 A ax ay - az 3 3 3 1 3 5 7 C ax ay az 4 4 4 4 1 1 17 19 3 ( A C) ax ay az 3 4 12 12 4 1 1 17 19 3 ( ay az) x ( ax ay az) 2 2 12 12 4 20 1 1 17 19 3 ( ay - az) x ( ax ay az) 2 2 12 12 4 21 A=5ax+3ay+2az , B=-ax+4ay+6az , C=8ax+2ay Find α and β such that : αA+ βB + C is parallel to y-axis αA+ βB + C= [ 5α ax+3α ay+ 2α az] + [-β ax+4β ay+6β az] +[8ax+2ay] =(5α- β+8)ax + (3α +4β +2)ay+ (2α +6β) az 5α- β+8=0 …..(1) 2α +6β=0 ……(2) solving (1) and (2) α=-1.5 , β=-0.5 22 A= αax+ 3ay- 2az , B=4ax+ β ay+ 8 az (a) Find α and β if A an B are parallel? AxB=|A||B|sin 0 =0 23 A= αax+ 3ay- 2az , B=4ax+ β ay+ 8 az (b) Relationship between α and β if A an B are perpendicular? A.B=|A||B| cos 90 =0 4α + 3β -16 = 0 α= 0.25 – 0.75 β 24 (b) Show that ay x az ax ax . ay x az ay x az ax ax . ay x az ax .ax ax ax ax ax 2 2 | ax || ax | cos 0 | ax | 1 25 (3) Scalar Triple Product A.(BxC)=B.(CxA)= C.(AxB) = volume of Parallelepiped متوازي السطوح 26 27 Show that A.(BxC)=(AxB).C 28 LOGO iugaza2010.blogspot.com melasmer@gmail.com 29